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Chapter 10: Problem 15

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \sin ^{k}\left(\frac{1}{k}\right) x^{k}$$

Short Answer

Expert verified
Answer: The radius of convergence is 1, and the interval of convergence is \((-1, 1)\).

Step by step solution

01

Applying the Ratio Test

Let's apply the Ratio Test to the given power series: $$\lim_{k\to\infty} \left| \frac{a_{k+1}}{a_k} \right|$$ where \(a_k = \sin^{k}\left(\frac{1}{k}\right) x^{k}\).
02

Calculate the Limit

Now, we calculate the limit: $$\lim_{k\to\infty} \left| \frac{\sin^{k+1}\left(\frac{1}{k+1}\right) x^{k+1}}{\sin^{k}\left(\frac{1}{k}\right) x^{k}} \right| = \lim_{k\to\infty} \left| \frac{\sin\left(\frac{1}{k+1}\right)}{\sin\left(\frac{1}{k}\right)}\right|^k \cdot |x| $$ By applying the Squeeze theorem, we know that \(\lim_{k\to\infty} \frac{\sin\left(\frac{1}{k+1}\right)}{\sin\left(\frac{1}{k}\right)} = 1\). So, we have: $$\lim_{k\to\infty} \left| \frac{\sin^{k+1}\left(\frac{1}{k+1}\right) x^{k+1}}{\sin^{k}\left(\frac{1}{k}\right) x^{k}} \right| = |x|$$
03

Compare to 1

For convergence using the Ratio Test, the limit of the ratio should be less than 1. In our case, we get: $$|x| < 1$$ Thus, the radius of convergence is 1.
04

Check Endpoints

We will now check the endpoints of the interval of convergence, which are -1 and 1, to see if they are included in the final interval. Case 1: When \(x = -1\) $$\sum \sin^{k}\left(\frac{1}{k}\right) (-1)^{k}$$ This series does not converge, as it does not meet the conditions for the Alternating Series Test. Case 2: When \(x = 1\) $$\sum \sin^{k}\left(\frac{1}{k}\right)$$ This series also does not converge, as it does not meet the conditions for convergence based on the Limit Comparison Test.
05

Determine the Interval of Convergence

Based on our analysis of the endpoints, the interval of convergence is \((-1, 1)\).

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Most popular questions from this chapter

The function \(\operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is called the sine integral function. a. Expand the integrand in a Taylor series about 0 . b. Integrate the series to find a Taylor series for Si. c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed \(10^{-3}\).

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{-3 x}$$

Teams \(A\) and \(B\) go into sudden death overtime after playing to a tie. The teams alternate possession of the ball and the first team to score wins. Each team has a \(\frac{1}{6}\) chance of scoring when it has the ball, with Team \(\mathrm{A}\) having the ball first. a. The probability that Team A ultimately wins is \(\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}\) Evaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is \(\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1} .\) Evaluate this series.

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty} e^{-k x}$$
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